The difference of convex algorithm on Riemannian manifolds

Abstract

In this paper we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. Equivalence between the classical and a simplified Riemannian version of the DCA is stablished. We also prove that under mild assumptions the Riemannian version of the DCA is well defined and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Some duality relations between the DC problem and its dual are also established. Applications to the problem of maximizing a convex function in a compact set and manifold-valued image denoising are presented in the DC context.